on-shell diagrams in sym beyond the planar limit...overview scattering amplitudes in planar n=4 sym...

On-shell diagrams in SYM beyond the planar limit

Brenda PenanteQueen Mary University of LondonHumboldt University of Berlin 12/01/16

Total Positivity: a bridge between Representation Theory and Physics

- Franco, Galloni, BP, Wen hep-th/1502.02034 based on:

- University of Kent -

Overview

Scattering amplitudes in planar N=4 SYM admit adescription in terms of the positive Grassmannian.

Loop leading singularities are residuesof a Grassmannian integral.

All-loop integrand determined via theBCFW recursion relation.Britto, Cachazo, Feng, Witten / Arkani-Hamed, Bourjaily, Cachazo, Caron-Huot, Trnka

Arkani-Hamed, Cachazo, Cheung, Kaplan / Mason, Skinner

... but how to go beyond planar N=4 SYM?

No ordering = No positivity

fixed , with Planar limit:

( Finite N corrections multiple traces )

Partial amplitude(colour ordered)

gauge group

Tree-level amplitudes enjoy Yangian symmetry

[[ Yangian = Superconformal + Dual Superconformal ]]Drummond, Henn, Plefka

Drummond, Henn, Korchemsky, Sokatchev

Loop level:Yangian symmetry broken due to IR divergencesLoop integrand

Rational function ofexternal and loop momenta

Planar loop integrand

dummy variables, but must be defined consistently among various terms

Planar loop integrandAmbiguities:

Planar loop integrand well defined: dual variables

State of the art in the planar limitAll-loop integrand determined by the all-loop recursion relation

Arkani-Hamed, Bourjaily, Cachazo, Caron-Huot, Trnka

Dual variables allow different terms in recursion relationto be combined in a non-ambiguous way

State of the art in the planar limit

Unavailable for non-planar integrands

??Non-planar integrand not well-defined

Can still study non planar Leading SingularitiesEden, Landshoff, Olive, Polkinghorne / Britto, Cachazo, Feng

All-loop integrand determined by the all-loop recursion relationArkani-Hamed, Bourjaily, Cachazo, Caron-Huot, Trnka

Dual variables allow different terms in recursion relationto be combined in a non-ambiguous way

"Cut" propagators:

Compute residue of the integrand Leading singularities: Maximal number of propagators cut (4xL)Ex: 1-loop

solutions

cut 4 propagators

Leading Singularities[[ Eden, Landshoff, Olive, Polkinghorne / Britto, Cachazo, Feng ]]

Planar

In the planar limit basis ofdual conformal integrands with "unit leading singularity"

LS are sufficient to determine the all-loop integrand!

All planar LS are residues of a(positive) Grassmannian integral

Positive Grassmannian parametrised by planar on-shell diagrams

[[ Arkani-Hamed, Bourjaily, Cachazo, Trnka - 2010 ]]

Non-planar integrand not well defined

Planar Non-Planar

Consider non-planar LS

Residues of a Grassmannian integral

Parametrised by non-planar on-shell diagrams

In the planar limit basis ofdual conformal integrands with "unit leading singularity"

LS are sufficient to determine the all-loop integrand!

All planar LS are residues of a(positive) Grassmannian integral

Positive Grassmannian parametrised by planar on-shell diagrams

[[ Arkani-Hamed, Bourjaily, Cachazo, Trnka - 2010 ]]

Motivation

Grassmannian formulation is linked toon-shell diagrams

Region of the Grasmmannian + dlog on-shell form

Arkani-Hamed, Bourjaily, Cachazo, Goncharov, Postnikov, Trnka

[[ Arkani-Hamed, Bourjaily, Cachazo, Trnka ]]Conjecture: Non-planar ampshave only log singularities and no poles at infinity.

Non-planar on-shell diagrams are the natural objects to study

Outline1. Grassmannian formulation for amplitudes

2. On-shell diagrams

3. Conclusions + more possible applications

Review of planarNon-planar corrections

Review of planarGeneralised face variablesGeneral boundary measurementA new type of singularityEquivalence, reductions, and polytopes

NotationSU(N) colour generators

Polarisation vectorsKinematics

Spinor variables

Partial amplitude:

(kinematics and polarisation)

Auxiliary fermionic variables(helicity)

(colour ordered)

Spinor products

Grassmannian Formulation

[[ Arkani-Hamed, Cachazo, Cheung, Kaplan - 2009 ]][[ Mason, Skinner - 2009 ]]

and a fermionic 4-plane :

The geometry of momentum conservation

Organise them in the , 2-planes in :

External data: for each particle

The geometry of momentum conservation

(Super) momentum conservation:

DEF: Grassmannian is the space of -planes in

Element of : choose -vectors:

gauge redundancy

Grassmannian formulation[[ Arkani-Hamed, Cachazo, Cheung, Kaplan - 2009 ]]

Scattering of gluons For us:: total number of gluons: number of gluons

Recall:

k=2k=n-2

MHVMHV

k=0, 1, n-1, n Amp = 0


Coordinates in Maximal minors (Plücker coords.)

Positive Grassmannian

Plücker relations: Ex:


Planar LS are residues of the following integral over

Gauge fix entries of




Ensure




Ensure

consecutive minors of

Ex:



Non-zero only if the plane is orthogonal to , and contains

Momentum conservation

Note:For impossible to have For impossible to have

Amplitudes automatically zero!




Non-planarPoles when consecutive minors vanish

invariance: cross ratio of minors

Ex: k=3

No notion of ordering or positivity in non-planar case

[[ Galloni, Franco, BP, Wen - 2015 ]]

On-shelldiagrams

[[ Arkani-Hamed, Bourjaily, Cachazo, Goncharov, Postnikov, Trnka - 2012 ]]

On-shell formulation[[ Arkani-Hamed, Bourjaily, Cachazo, Goncharov, Postnikov, Trnka - 2012 ]]

Bi-coloured graphs made of the building blocks:

Edges:

Nodes

on-shell momentum

MHV amplitude

MHV amplitude

Constructing on-shell diagramsTo connect two nodes, integrate over on-shell phase spaceof edge in common:

Little group

Can construct more complicated diagramsEach node has two degrees of freedom

Constructing on-shell diagramsExamples:

13

2

4 5

13

54

2

1

2

5

43

4

1 2

3


Planar:Can be embedded on a disk

13

2

4 5

13

54

2

1

2

5

43

4

1 2

3


Planar:Can be embedded on a disk

Non planar:Can be embedded on a surface withmultiple boundaries/ higher genus13

54

2

1

2

5

43

13

2

4 5

4

1 2

3

Equivalence MovesOn-shell diagrams are equivalent if they are related by thefollowing moves:Merger:

Square move:

Note that every on-shell diagram can be made bipartiteNote that every on-shell diagram can be made bipartite

Fusing Grassmannians

On-shell diagram

Boundary measurement

Ex: 4

12

3

An on-shell diagram with black nodes, white nodesand internal edges is associated to , where:

Bipartite technology

Perfect matching Choice of edges such that every internal node is the endpoint of only one edge

Perfect orientation

4

12

34

12

34

12

3

Bipartite technology

Perfect matching Choice of edges such that every internal node is the endpoint of only one edge

Perfect orientation

4

12

34

12

34

12

3

Sources

Sinks

4

12

3 4

12

34

12

34

12

34

12

34

12

3 4

12

3

Example4

12

3

Perfect matchings:

4

12

3 4

12

34

12

34

12

34

12

34

12

3 4

12

3

Oriented perfect matchings:

Example4

12

3

4

12

3 4

12

34

12

34

12

34

12

34

12

3 4

12

3

Oriented perfect matchings:

Example4

12

3

Flows:4

12

3 4

12

34

12

34

12

34

12

34

12

3 4

12

3

Reference


4

12

3 4

12

34

12

34

12

34

12

34

12

3 4

12

3

Flows:

Map between on-shell diagram and element of the Grassmannian

Reference

Flows from to 41 2 31

2 Sign prescription

4

12

3


Plücker coordinates are positive in planar case and are a sum of flows with corresponding source set.

Ex:

41 2 3

1

2

4

12

3 4

12

34

12

34

12

34

12

34

12

3 4

12

3

Sign prescription

ImportantOn-shell diagrams parametrise regions of the Grassmannian

4

12

3

Equivalent diagrams parametrise the same region:

4

12

3

Graph is reducible if possible to delete edges while preserving region.4

12

3

Otherwise graph is reduced.This changes for non-planar graphs!Planar: set of non-zero minors preserved

Parametrising on-shell diagramsPlanar:

# degrees of freedom of a planar on-shell diagram is

# faces

4

12

3

Face variables: ?General for non-planar

4

12

3

Edge variables:

On-shell dlog form: variables unfixed by delta-functions mapped to loop integration variables.

Bases for expressing flows: Edges and Faces

Generalised face variables

Faces

Paths connecting different boundaries

Fundamental cycles

1

3

2

4

6

5

1

3

2

4

6

5

1

3

2

4

6

5

α β b

Ex: Genus 1

F = # facesB = # boundariesg = genus

[[ Galloni, Franco, BP, Wen - 2015 ]]

Faces

Paths connecting different boundaries

Fundamental cycles

dlog on-shell form:

F = # facesB = # boundariesg = genus

Generalised face variables[[ Galloni, Franco, BP, Wen - 2015 ]]

Some properties of planar graphs

If it is impossible to remove an edge of a graph without sending some Plücker coord to zero, the graph is reduced.

(Positroid stratification of )

[[ Arkani-Hamed, Bourjaily, Cachazo, Goncharov, Postnikov, Trnka - 2012 ]]

Boundaries of the regions in the positive Grassmannian are always

Recall:

A non-planar novelty:

2

3

4

56

1

It is possible to remove an edge of a reduced graph withoutsending any Plücker coord to zero!

Recall: Deformation from planar Grassmannian integrand

Non-planar novelties[[ Arkani-Hamed, Bourjaily, Cachazo, Postnikov, Trnka - 2014, Galloni, Franco, BP, Wen - 2015 ]]

Method for determining : generalisation of [[ Arkani-Hamed, Bourjaily, Cachazo, Postnikov, Trnka - 2014 ]]

from k=2 leading singularities to higher kOBS:

Removal of an edge does not set anyto zero, but gives rise to the relation

2

3

4

56

1

A non-planar novelty:It is possible to remove an edge of a reduced graph withoutsending any Plücker coord to zero!


Reducibility & Equivalence: Non-planar[[ Arkani-Hamed, Bourjaily, Cachazo, Postnikov, Trnka - 2014, Galloni, Franco, BP, Wen - 2015 ]]

2

3

4

56

1

A non-planar novelty:It is possible to remove an edge of a reduced graph withoutsending any Plücker coord to zero!


Reducibility & Equivalence: Non-planar[[ Arkani-Hamed, Bourjaily, Cachazo, Postnikov, Trnka - 2014, Galloni, Franco, BP, Wen - 2015 ]]

Polytopes

Sum of flows with source set with coefficients

[[ Gekhtman, Shapiro, Vainshtein - 2013 ]] Annulus[[ Franco, Galloni, Mariotti - 2013 ]] Arbitrary B, genus zero[[ Franco, Galloni, BP, Wen - 2015 ]] Any graph

Point in matching polytope

Point in matroid polytope

Notions of equivalence/reduction can be rephrased in terms of polytopes:

Plücker coord.

[[ Postnikov, Speyer, Williams - 2008, Franco, Galloni, Mariotti - 2013 ]]

Non-vanishing minors and relations can be seen from the polytopes

on-shell diagram

Perfect matching

Perfect matchings with same source set

Characterisation of on-shell diagrams

Equivalent graphs have the same matroid polytope

A graph is reduced if it is impossible to remove edges while preserving the matroid polytope.

Cell of the Grassmannian characterised by a zig zag path

Non-planar without extra constraintson Plücker coordinates

planar / reduced

Cell of the Grassmannian characterised by matroid polytope

Reduction of a diagram removes 1 dof while preserving the matroid polytope

Constraints and polytopes: example[[ Galloni, Franco, BP, Wen - 2015 ]]

2

3

4

56

1

Before removal: 40 perfect matchings

Before and after removal: After removal disappears

After removal: 33 perfect matchings

Concluding remarks & Outlook1) Physical interpretation:

All tree level amplitudes and loop integrandsvia BCFW recursion relation.

Planar:

dlog form of the loop integrand:


Britto, Cachazo, Feng, Witten / Arkani-Hamed, Bourjaily, Cachazo, Caron-Huot, Trnka

Concluding remarks & Outlook

Leading singularities of the loop integrandNon planar:Non-planar loop integrand?

[[ Arkani-Hamed, Bourjaily, Cachazo, Trnka - 2014 ]]Conjecture: Non-planar ampshave only log singularities and no poles at infinity.

Non-planar Grassmannian formulation?

1) Physical interpretation:All tree level amplitudes and loop integrandsvia BCFW recursion relation.

Planar:

dlog form of the loop integrand:


Britto, Cachazo, Feng, Witten / Arkani-Hamed, Bourjaily, Cachazo, Caron-Huot, Trnka


2) Non-planar diagrams parametrise regions of withhidden relations between Plücker coordinates.

? Method for finding representative graph given a constraint




3) MHV non-planar leading singularities are sums of planar ones.

Same not true for Non-MHV, however similar method can beused to find the deformation of the integrand .

[[ Arkani-Hamed, Bourjaily, Cachazo, Postnikov, Trnka - 2014 ]]




3) MHV non-planar leading singularities are sums of planar ones.

Same not true for Non-MHV, however similar method can beused to find the deformation of the integrand .

[[ Arkani-Hamed, Bourjaily, Cachazo, Postnikov, Trnka - 2014 ]]

4) Positive Grassmannian Non-planar generalisation?

Amplituhedron[[ Arkani-Hamed, Trnka - 2013 ]]

[[ Evidence: Bern, Herrmann, Litsey, Stankowicz, Trnka - 2015 ]]

on-shell diagrams in sym beyond the planar limit...overview scattering amplitudes in planar n=4 sym...

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